In this paper, we study the iterations of quasi $phi$-nonexpansive mappings and its applications in Banach spaces. At the first, we prove strong convergence of the sequence generated by the hybrid proximal point method to a common fixed point of a family of quasi $phi$-nonexpansive mappings.  Then, we give  applications of our main results in equilibrium problems.

The purpose of this paper is to introduce two implicit iteration schemes for approximating fixed points of nonexpansive mapping T and a finite family of nonexpansive mappings {Ti}i 1, respectively, in Banach spaces and to prove weak and strong convergence theorems. The results presented in this paper improve and extend the corresponding ones of H.-K. Xu and R. Ori, 2001, Z. Opial, 1967, and oth...

We propose a strongly convergent algorithm for finding a common point in the solution set of a class of pseudomonotone equilibrium problems and the set of fixed points of nonexpansive mappings in a real Hilbert space. The proposed algorithm uses only one projection and does not require any Lipschitz condition for the bifunctions. AMS 2010 Mathematics subject classification: 65 K10, 65 K15, 90 C...

It is shown that every asymptotically regular or λ-firmly nonexpansive mapping T : C → C has a fixed point whenever C is a finite union of nonempty weakly compact convex subsets of a Banach spaceX which is uniformly convex in every direction. Furthermore, if {Ti}i∈I is any compatible family of strongly nonexpansive self-mappings on such a C and the graphs of Ti, i∈ I, have a nonempty intersecti...

In this paper we prove the weak and strong convergence of the implicit iterative process with errors to a common fixed point of a finite family {Tj}i=1 of asymptotically quasi Ij−nonexpansive mappings as well as a family of {Ij}j=1 of asymptotically quasi nonexpansive mappings in the framework of Banach spaces. The obtained results improve and generalize the corresponding results in the existin...

We discuss the equilibrium problem for a continuous bifunction over the fixed point set of a firmly nonexpansive mapping. We then present an iterative algorithm, which uses the firmly nonexpansive mapping at each iteration, for solving the problem. The algorithm is quite simple and it does not require monotonicity and Lipschitz-type condition on the equilibrium function. At the end of the paper...

where F is a monotone operator. Recently, Lu et al. [] were concerned with a special class of variational inequalities in which the mapping F is the complement of a nonexpansive mapping and the constraint set is the set of fixed points of another nonexpansive mapping. Namely, they considered the following type of monotone variational inequality (VI) problem: Find x∗ ∈ Fix(T), such that 〈(I –V ...